Observer-Based Output-Feedback Synchronization of Fractional-Order Fuzzy Clifford-Valued Neural Networks: A Cascade Certificate and the Quantified Price of Output Feedback

Abstract

Synchronization certificates for fractional-order neural networks often assume the full multivector state is measured. In practice only a partial output is available—the full multivector states of a subset of neurons—and the controller must run on an estimate. We give an output-feedback theory for leader-following synchronization of fractional-order Takagi–Sugeno fuzzy Clifford-valued neural networks in which each follower measures only some of its neurons, runs a local Luenberger-type fractional observer to reconstruct the rest, and pins the estimate to the leader (certainty equivalence). The analysis has three parts. First, a detectability certificate: a per-neuron M-matrix test on the observer-error energies under which the observer error decays, in which the output-injection gain raises the damping of the measured neurons only; the gain is essential for feasibility—without it the certificate is infeasible (\(a_o = (-2.34, -2.15, 1.67)\) vs. \((2.86, 3.06, 1.67)\))—and the simulator shows the open-loop error failing to synchronize (stalling at 0.80) while the gain drives it to \(2.6 \times 10^{-4}\). Second, a cascade certificate: the observer error is autonomous and drives the tracking error through a coupling \(\theta^{-1}\|H\|^2\); a triangular \(2\times 2\) fractional comparison gives joint synchronization at the rate \(\min(\text{observer}, \text{tracking})\), which is observer-limited here (1.27). Third, the price of output feedback, made quantitative: the coupling forces a tracking-rate loss \(\theta\) (the “separation” is weak, not clean), and the residual is \(14\times\) larger than full-state feedback; under a bounded sensor attack both ultimate errors scale as \(\bar{d}^2\), amplified by the control gain \(\|H\|^2\)—so high-gain pinning that speeds tracking also magnifies sensor-attack sensitivity. Selected inequalities and certificates are numerically audited, and the associated certified–empirical gap is measured and reported.